Stack Exchange Network

Stack Exchange network consists of 174 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

Questions tagged [topology]

In mathematics, topology examines the properties of space (such as connectedness and compactness) that are preserved under continuous deformations, such as stretching and bending, but not tearing or gluing.

0
votes
0answers
29 views

Can velocity wrap around?

Thermodynamics often plays out in $2n$ dimensional space. It is convenient for some numerical simulations to treat this space as if it was toroidal, wrapping around in every dimension (the fast ...
1
vote
0answers
50 views

Pure math courses for physicists: Topology [on hold]

I'm in my bachelor in physics. In a couple of weeks I start my last year, and I'm interested in taking some pure math courses. As you see, I like the theoretical point of view, but I don't know if the ...
0
votes
0answers
11 views

A good instruction on Symmetry enriched Topological phases

I am looking for a good introduction to SETs, and topologically ordered phaeses it should be something describing first principles and gives a good explanation on the basics and the logic of this ...
0
votes
1answer
61 views

Reflection Vector (Ray Tracing)

Snell's law of refraction at the interface between 2 isotropic media is given by the equation: \begin{align} \tag{1} n_1 \,\text{sin} \,\theta_1 = n_2 \, \text{sin}\,\theta_2 \end{align} $\qquad$ ...
0
votes
0answers
16 views

2D BHZ tight binding model for Quantum spin Hall insulator

I am currently reading this article : https://arxiv.org/abs/cond-mat/0611341 and want to derive the k-space tight binding model of 2D BHZ. The tight binding model is written as \begin{equation} H = \...
1
vote
1answer
37 views

Is it possible understand Berry curvature as Gaussian curvature in some limit?

I would like to understand the Berry curvature and the Chern number from mathematical geometry-topology. I understand that in electronic QHE, there is a map from $k^2$ to a vector space where the ...
6
votes
1answer
128 views

Spacetime has an infinite number of choices for differentiability. Coincidence?

Spacetime can be modelled using a four-dimensional topological manifold. Say we denote the manifold using $(M, \mathcal{O})$ where $\dim M =d$. The structure $(M,\mathcal{O})$ is not sufficient for ...
1
vote
1answer
95 views

How is Inflation able to create an infinite amount of energy?

Follow up to this question here: If the universe is flat, does that imply that the Big Bang produced an infinite amount of energy? As I understand Inflation theory, some time after the Big Bang, the ...
0
votes
2answers
31 views

Clarification on statement in “Unitary Symmetry and Elementary Particles” by Lichtenberg

He says that: The set of values of the parameter or parameters which characterize a group element can be considered to be points in some kind of space. The number of parameters characterizes the ...
2
votes
0answers
61 views

About Witten's path integral formulation of Jones polynomial

In his landmark paper Quantum field theory and the Jones polynomial, Witten proposed that the Jones polynomial can be obtained by the expectation value of the Wilson loop operators over links in the ...
0
votes
0answers
16 views

Are fractional quantum hall effect system symetry enriched topological phases?

In the papers I review they first start to talk about topologically ordered phases of matter. Their standard example of it is FQHE. Than they give another set examples which are quantum spin liquids, ...
6
votes
1answer
64 views

2D global conformal transformations and the $z= \frac1w$ argument

For instance in Blumenhagen's CFT, there is a standard argument which determines that globally defined conformal transformations on the Riemann sphere where $$l_n = -z^{n+1} \partial_z$$ is an ...
1
vote
1answer
33 views

Is the quantum Hall state a topological insulating state?

I am confused about the quantum Hall state and topological insulating states. Following are the points (according to my naive understanding of this field) which confuse me: Topological insulator has ...
2
votes
1answer
23 views

Are interacting symmetry protected topological (SPT) phases and symmetry enriched topological (SET) phases must be gapped?

I wonder are interacting SPT and SET phases gaped? Can we have a SET or interacting SPT phase in a semi metal?
1
vote
1answer
50 views

Poincaré and Galilei group - notation

On this slide it just says that $\mathcal{P}$ and $\mathcal{G}$ are the Poincoré and Galilei groups, but I do not understand what they are made of. What does $\mathbb{R}^{1,3}$ mean? Why does $\...
5
votes
2answers
89 views

Spin statistics from the fundamental group of $SO(D)$

I read the answer to this question and am very intrigued by its simple and elegant explanation of the emergence of anyon, boson & fermion statistics. @Trimok basically says: In a space-time ...
1
vote
1answer
78 views

Rigorous definition of generalized coordinates

In Goldstein's classical mechanics and in many other books I haven't seen a rigorous definition of generalized coordinates. In a system of $N$ particles described by $\textbf{r}_1, \dots, \textbf{r}...
2
votes
1answer
35 views

Configuration space of identical particles - fractional statistics

In Khare's book of fractional statistics and quantum theory, when discussing why we need fractional statistics he arrives at the configuration space for a system of two identical particles in $d$ ...
2
votes
1answer
70 views

Chern-Simons and framing dependence$.$

According to ref.1, the correlation functions of a Chern-Simons theory are topological invariants, up to the so-called framing, that is, the trivialisation of $TM\oplus TM$. The origin of this framing ...
3
votes
2answers
112 views

On the topology of the Einstein-Rosen bridge

I am currently working with Harvey Reall's notes on Black holes, and I have 2 questions concerning the interpretation of the Einstein-Rosen bridge. After some algebra, we are faced with the metric: $$...
5
votes
1answer
68 views

How does extending a Chern-Simons theory to the bulk fix potential singularities?

According to ref.1 (§A.3), the naive definition of Chern-Simons $$ S[A]=k\int_M \mathrm{CS}[A]\tag{A.17} $$ is ill-defined, because $A$ may have "Dirac string singularities". The solution is to extend ...
1
vote
1answer
31 views

Must the electromagnetic 2-form be harmonic in vacuum?

The Maxwell equations in vacuum are $dF=0$ and $d*F=0$. Is this not the same as saying $F$ is both closed and co-closed, and hence harmonic? But Hodge's theorem says the space of harmonic $p$-forms on ...
1
vote
1answer
52 views

Recipe to determine symmetries of quadratic fermionic Hamiltonian in second quantisation

Consider an arbitrary 1D chain (of length $N$) of fermions with an arbitrary quadratic Hamiltonian of the form $$\mathcal{H}=\hat{\Psi}^\dagger H \hat{\Psi}$$ with $$\hat{\Psi}=\left(a_1, a_2, ...,...
4
votes
1answer
312 views

Can solutions of GR have non-zero genus?

Imagine there is "cavity" in one's locally Lorentzian manifold (the manifold has non-zero genus). Have these kind of solutions in general relativity been considered? If so, where can I read more about ...
6
votes
5answers
176 views

Nature of the elements of spacetime?

I am learning about relativity and am not quite sure how to think of spacetime. From a mathematical perspective, spacetime is a manifold i.e. a topological space for which about any point there exists ...
1
vote
1answer
78 views

Why are two different gauge transformations of $A_\mu=0$ in $U(1)$ gauge thoery equivalent?

Two inequivalent gauge transformations of $\mathbb{A}_\mu=0$, described by $U$ and $\tilde{U}$ of a pure $SU(N)$ Yang-Mills theory as $$\mathbb{A}_\mu=\frac{i}{g} U\partial_\mu U^\dagger~\text{and}~\...
3
votes
3answers
127 views

Topological indices for systems that lack translational invariance

I have a 1D discrete, finite system that lacks translational invariance. It appears to have edge states, in much the same way as an SSH model has edge states. In the SSH model we can study the ...
1
vote
0answers
58 views

Can one find Dirac matrices for any spacetime metric?

For any metric $$g_{μν}$$ is there always a linearly independant spacetime algebra satisfying $$\{γ_μ,γ_ν\} = 2 g_{μν} I?$$ For a diagonal metric I was able to work out that $$\bar{γ}_μ=\sqrt{n_{μμ}*...
2
votes
2answers
47 views

Is the classification of (Symetry Protected) Topological Order for 3 band models different than for two band models?

I was reading this article: https://arxiv.org/abs/1512.08882 on the 10 fold way which gives a nice explanation of the possible topological phases for each of the symmetry classes. The example ...
5
votes
1answer
64 views

What is the topology of a phase diagram?

Looking at various two-variable phase diagrams I was struck by that on every one I have seen so far all the phases formed simple connected regions; see, for example the phase diagrams of $H_2O$ or of $...
0
votes
1answer
78 views

What is direct interaction, if exist, between gluons and pions?

Gluons mediate the strong force between quarks. Pions mediate the nuclear force or nucleon-nucleon interaction or residual strong force. I had thought of some scalar bosons for my idea because if I'm ...
1
vote
1answer
70 views

If the universe is a $3$-torus, must it be longer in some direction?

So in a finite universe, I've read one possible topology for a flat universe is a 3 torus. On a 2 torus, it's obviously longer in one direction than the other. Would the same hold true for the ...
2
votes
0answers
44 views

Conformal compactification of spacetimes

I’m looking answers and/or references concerning the following questions about conformal compactification. Given a $d$-dimensional spacetime $(M,g)$ (or simply just for $d=4$): Does the conformal ...
6
votes
1answer
327 views

Do there exist phases of matter where the order parameter space is non-orientable?

For example, are there order parameter space that is homeomorphic to a Klein bottle?
4
votes
1answer
122 views

Are Minkowski and Schwarzschild spacetimes diffeomorphic?

Another mathematical question, arising from GR. Some days ago I wrote, in an answer to 1, that they are. Then @magma commented they are not. He promised a proof, but none appeared. After magma's ...
1
vote
2answers
67 views

vortex anti-vortex configuration to ground state [closed]

i'm studying kosterlitz transition. I'm reading this: https://assets.nobelprize.org/uploads/2018/06/advanced-physicsprize2016-1.pdf?_ga=2.51324009.1302372948.1538119052-1605759177.1538119052 Now ...
2
votes
1answer
145 views

Vortex anti-vortex?

i'm studying Kosterlitz THouless transition and i have a doubt: what is a vortex anti-vortex configuration? Is this thing? or this one I think that they are quite different !
3
votes
3answers
146 views

Kruskal-Szekeres Coordinates and the Singularity

On the wikipedia page for Kruskal-Szekeres coordinates, it states: [Kruskal-Szekeres] coordinates have the advantage that they cover the entire spacetime manifold of the maximally extended ...
1
vote
1answer
66 views

Winding number and area formula

I need to show some properties of the topological expression involving a map $\vec{n}(x): S^2 \rightarrow S^2$ $$W=\frac{1}{4\pi}\int \vec{n} \cdot (d\vec{n} \wedge d\vec{n}),$$ but I am not very ...
3
votes
2answers
77 views

Global conformal group in 2D Euclidean space

This is a rather naive question, but I was just wondering. I know that the local conformal algebra of 2d Euclidean space is the direct sum \begin{equation} \cal{L}_0\oplus\overline{\cal{L}_0}, \end{...
1
vote
0answers
38 views

What is this metric's scale factor?

While answering this question about a hypothetical 3-sphere universe $S^3$ expanding with a constant acceleration $\phi$ from a zero initial speed $$ r=\dfrac{\phi}{2}t^2$$ I started from a generic ...
0
votes
0answers
28 views

Visualizing k-space tori in 3D

In many introductions to topological insulators (in the exposition of Haldane’s model, for example), we represent the parameter space, a torus, on a plane with axes running from $0$ to $2\pi$. In an ...
0
votes
3answers
61 views

Gaussian surface and closed surfaces

What is (intuitively speaking) a closed surface? (this question may seem trivial but I think it's not so clear w.r.t. topology. Some definitions on Wikipedia seem confusing. It is said that a sphere ...
2
votes
1answer
89 views

When are we required to use the Wess-Zumino term?

I was recently reading about non-Abelian bosonization, and I had a question concerning the Wess-Zumino term. In particular, I have been reading this short introduction by Ivan Karmazin, which states ...
2
votes
1answer
165 views

How can I trace the path of a photon on space-time defined by this metric?

I have a hypothetical universe where the relation between space and time is defined by this metric: $$ds^2=-\phi^2dt^4+dx^2+dy^2+dz^2$$ Where $\phi = 1.94 \times 10^{-14}\space km\space s^{-2}$ (note ...
0
votes
1answer
27 views

Does 1+1D Gravity Contract to a point if the spatial dimension has the topology of a circle?

I image if I put a point mass on a circle, a test particle anywhere on the circle will move toward it. I am now trying to imagine the shape of space-time for this by embedding it into a 3D space. I ...
4
votes
1answer
118 views

Do higher homotopy groups play any role in gauge theory?

As is more-or-less well-known, the magnetic monopoles of a gauge theory are classified by the first homotopy group of the gauge group, $\pi_1(G)$ (cf. Lubkin (1963)). The second homotopy group is ...
4
votes
1answer
76 views

What justifies compactifying space and spacetime, in the context of instantons?

When studying Yang-Mills instantons, there are two instances where one compactifies a space. When classifying vacuum states, one demands $A_\mu(\mathbf{x})$ to become a constant as $\mathbf{x} \to \...
2
votes
2answers
84 views

Is there a physical interpretation of symplectic manifolds which are not cotangent bundles?

The inspiration for symplectic geometry was from Hamiltonian mechanics. However, I am wondering how close the ties are between arbitrary symplectic manifolds and real physical systems. In ...
0
votes
0answers
57 views

Does topological T-duality imply T-duality?

Topological T-duality was introduced in [1] for $S^1$ principal bundles and for $\mathbb{T}^n$ principal bundles in [2]. It seems to be motivated by the usual string theory T-duality by which I mean ...